Bob Monahon’s Bloghttps://bobmonahon.wordpress.com
Science, Technology, Economics and Politics. Broad enough scope, eh?Thu, 09 Sep 2010 01:45:34 +0000enhourly1http://wordpress.com/https://s2.wp.com/i/buttonw-com.pngBob Monahon’s Bloghttps://bobmonahon.wordpress.com
Quantum Mechanics in a nutshellhttps://bobmonahon.wordpress.com/2010/08/29/quantum-mechanics-in-a-nutshell/
https://bobmonahon.wordpress.com/2010/08/29/quantum-mechanics-in-a-nutshell/#respondSun, 29 Aug 2010 21:25:04 +0000http://bobmonahon.wordpress.com/?p=55This takes a little time to get to the point of the main question:

How come we can’t say that a particle is *right there* and that it is moving *that fast* in *that direction* ?

So bear with me. I’m going through these steps:

1. The electron is a particle.

2. The electron acts like a particle

3. Sometimes the electron doesn’t act like a particle

4. Sometimes the electron acts like a wave

5. What is the location of the electron in QM?

6. If the QM location is a probability distribution, what is the QM velocity?

7. The location and velocity are both imprecise in QM.

1. The electron is a particle

In a very basic sense, QM is a math theory that describes what happens when electrons interact with matter – that is, with atoms. For instance, when you send a stream of electrons from an electron gun (which is actually at the back of all your CRT computer monitors), the electron goes through a vacuum and hits some target. In the CRT [Cathode Ray Tube], the target is the display screen. Precise experiments show that each electron hits in a single spot. Additional experiments show that the electrons are all of the same mass, and they all carry the same charge. So, they are particles. Nothing new there – that’s what we expected.

2. The electron acts like a particle

Then we come to the two-hole experiment. Here we shoot electrons from the electron gun through a partition that has two holes in it, and then measure the electrons when they hit a CRT screen on the far side. Here’s a really lame diagram that shows the experimental setup:

Diagram 2.1
Well, let’s work up to this experiment in steps. First, we’ll only keep the top hole open. When we do that, we find that the electrons hit the CRT screen mostly at the level of the top hole. However, some of them are a little higher, some of them are a little lower. If you measure the # of electrons at each point on the CRT screen, you get something like this (the dashed lines on the CRT screen show how many electrons you measure at each spot):

Diagram 2.2

On the CRT screen, you’ll see a fuzzy dot, brighter in the middle and fading towards the edges. This is easy to understand. Think of the electrons coming out of the gun and going through the hole as though they were a stream of bullets shooting out of a machine gun. The further the stream of bullets gets from the gun, the more it spreads out. The stream of bullets spreads because some bullets move slightly faster and slightly off-course from the center. The electron beam spreads for the same reason. Some shoot a little high, some shoot a little low, but most shoot right towards the center.

Now, repeat the experiment with only the bottom hole open, and you get this:

Diagram 2.3

Again, we have the same pattern – a fuzzy dot on the screen – but this time it is centered down at the level of the bottom hole.

So, we have a lot of experiments that show that the electron is a particle, and we have two experiments that show what these particles do when they go through a hole in a partition. So … what happens if we open both holes?

IF these electrons act like the particles we are used to, the CRT screen pattern should be the simple addition of the two one-hole patterns, like this:

Diagram 2.4

If we shoot two machine guns at a wall, the wall gets more bullet holes where both guns hit. If the electrons act like that, the CRT screen would show a larger fuzzy dot, with a bright spot smeared between the center of the top and bottom hole.

3. Sometimes the electron doesn’t act like a particle:

But when we run the experiment, we don’t get the simple addition pattern! Instead, we get this crazy pattern:

Diagram 3.1

This diagram is trying to show that there is a bright spot in the middle of the CRT screen, with two very dim spots right next to it, and then alternating bright and dim spots as you move further from the center of the CRT screen. As you go further from the center, the more distant bright spots are dimmer than the center spot, until the whole pattern dies out far from the center of the CRT.

What’s up with that? At some points on the screen, the patterns add up, and at other points they cancel each other out. Richard Feynman was awarded a Nobel Prize in physics, and he said that this one experiment defines the essence of the quantum mystery. I’ll go with his evaluation, since he’s the one with the Nobel Prize, not me.

4. Sometimes the electron acts like a wave

The experimental results above are perfectly natural in another part of physics – wave physics. For a moment, let the diagram above represent the surface of a swimming pool, with someone making waves on the far left. The surface waves go through the two holes in the partition, and then meet at the far end of the pool where we have the CRT screen. At some points on the screen, a wave crest from the top hole meets a wave crest from the bottom hole. The resulting wave is higher than either incoming wave: the wave heights add up, and that’s one of the bright spots. At other points on the screen, a wave crest from one hole meets a trough from the other. The wave heights cancel each other out, and that’s one of the dim spots. At a point where a trough from one meets a trough from the other, the wave depths add up: the trough at that point is deeper that either incoming trough. If the screen measures crests and troughs the same way (or if the screen somehow measures the SQUARE of the wave height, with depth being a negative height), the wave heights make a pattern on the pool wall just like the pattern we actually measure for electron density on the CRT screen.

Well, that right there is the quantum mystery: The electron, when it is measured at the screen, is a particle. The motion of the electron from the electron gun through the holes to the screen is the same as a wave. In QM, this wave is the “probability amplitude”, and the actual probability of the electron hitting the screen at any point is the probability amplitude squared.

OK, now we have an equation that explains the experimental result. It’s called the Schrödinger wave equation. It’s way too advanced for this little note. However, standard physics theory states that the electron can be found, probably, where the wave equation says it will be, and that’s all the theory states. But the theory works really well. For example, it predicts (probably) where the electron is located in the hydrogen atom, and how much energy it has when it leaves the hydrogen atom, and what energy light it emits and absorbs as it moves around in the hydrogen atom. The Dirac equation is an extension of the Schrödinger equation that accounts for special relativity (the electron can’t go faster than the speed of light). The Dirac equation explains all of the above plus something called the “spin” of the electron. So the math equations do explain the experimental results.

Finally I’m getting to your question. So what’s all this about “can’t find the position and velocity at the same time?”

5. What is the location of the electron in QM?

The location of the electron, in QM, is the square of the probability amplitude. That’s all the theory tells us. Of course, when the electron hits the screen, we know exactly (pretty close) where it is then.

Until we actually measure the electron, however, QM theory says the electron is located probably at the most likely spot, but maybe a little left or right or ahead or behind of the most likely spot. The QM location is a probability curve that looks like the bell-shaped curve. Here are a couple of examples (consider the electron is moving toward us, and the bell-shaped curve measures the probability that the electron is really a little left or right of the most likely position):

Diagam 5.1 – A wide bell-shaped curve.

Diagram 5.2 – A narrow bell-shaped curve:

The wide bell-shaped curve represents a particle where we expect a big dim fuzzy spot on the CRT screen. The electron has a pretty good chance of being far off-center. The narrow bell-shaped curve represents a particle where we expect a small bright fuzzy spot. The electron is more likely to be close to the center.

The QM equations always explain experimental results on the basis of probability. You can run any experiment with many electrons (or you can run it with only one electron at a time, but run it many times). In all cases, the count and distribution of electrons on a CRT screen will match the probability profile predicted by the QM equation.

6. If the QM location is a probability distribution, what is the QM velocity?

Instead of velocity, QM actually talks about momentum. (Momentum is simply mass times velocity.) The location and the momentum, in QM, are “conjugate variables”. In one sense, this means that location and momentum are enough to specify the motion of the particle. In another sense, this means that the location and momentum are tightly bound up in the equation of motion for the particle.

There’s nothing really new here. In traditional mechanics, for example, if you have the original location of a particle, and you know its initial velocity and its mass, you can calculate its trajectory. All the high-falutin’ words like “conjugate variables” just mean the same thing as that.

(Oh yes, to calculate the trajectory, you also have to know the force on the particle. You need to know the force for QM, too.)

Now, when the math tells us that the location is a bell-shaped curve, then it also says that the momentum is a bell-shaped curve. It turns out that these two curves are related by something called a Fourier Transform. Given the location curve, you calculate a Fourier Transform and you get the momentum curve.

Finally: The narrower one curve is, the wider the other curve is. That’s just how Fourier Transforms work.

7. The location and velocity are both imprecise in QM.

If we accept that QM explains everything about an electron, then the location at any time is not precise – it is a bell-shaped curve around the most likely position. Similarly, the momentum is also not precise – it is a (related) bell-shaped curve around the most likely momentum. And, since the curves are Fourier Transforms of each other, the narrower one is, the wider the other one is. This relationship between location and momentum is called the Heisenberg Uncertainty Principle. It states that you cannot know both the QM position and QM momentum exactly. In fact, it says more than that. It says you cannot know either variable exactly, and that the more precisely you know one, the less precisely you know the other.

Additional experiments have been made, and they all confirm the equations of Quantum Mechanics. How can this be, when we can actually measure the position of the electron on the CRT screen, and we know that the electron is particle with a very specific mass and electric charge? That’s the quantum mystery, straight and simple.

Physicists fall into (roughly) 3 groups: Group #1 have tried to develop actual physical models of how the electron moves, with an actual trajectory with location and momentum at all times specified. They’ve tied themselves in knots, and come up with some theories, but they are considered to be on the fringe of physics. Group #2 take a neutral stance and say that if the equations work, use them — and don’t ask the equations for an explanation of what’s actually happening. Group #3 take the stance that the electron doesn’t have a well-defined position or momentum, and you can’t even ask the question — the only thing you can ever know is the two probability curves.

Well, that’s it. How is it possible? No one really knows. But now, at least you know something about the experiments and the math that lie behind the statement:

“Quantum theory tells us that we have to abandon the familiar image of a particle (such as an electron or a photon) as having, at any time, a definite position or a definite velocity. […] This is not simply a matter of needing more precise measurements. The uncertainty is a fundamental aspect of physical reality.”

The writer of that quote is part of physicist group #3.

]]>https://bobmonahon.wordpress.com/2010/08/29/quantum-mechanics-in-a-nutshell/feed/0Bob MonahonTwo-hole experimentTwo Holes - bottom closedTwo Holes - top closedTwo Holes - Patterns addTwo Holes - Patterns interfereWide bell-shaped CurveNarrow bell-shaped curveThoughts on Earth Hour and the population of the worldhttps://bobmonahon.wordpress.com/2009/03/28/thoughts-on-earth-hour-and-the-population-of-the-world/
https://bobmonahon.wordpress.com/2009/03/28/thoughts-on-earth-hour-and-the-population-of-the-world/#respondSat, 28 Mar 2009 22:01:12 +0000http://bobmonahon.wordpress.com/?p=41So I’m reading comments on the web about Earth Hour – the annual international event to raise awareness of climate change. Sponsored by the WWF, the organization asks that everyone in the world to turn off their lights for one hour, beginning at 8:30 p.m. local time on Saturday the 28th of March, 2009. Today.

I followed a series of responses to a blog at the Christian Science Monitor, here:

Really, most of the response posts were heavily opinionated, sarcastic, and denigrating. The posts were from both liberal and conservative points of view — most were pretty bad, basically reverting to calling each other names. I love the level of intelligent discourse on the web…

So I took a side-track, and followed some posts about the population of the world. Why not?

Response post #1: “What good will this do? The reason we are so messed up is overpopulation …”

Later response post: “… you need to do some research. The world is not overpopulated. There is plenty of room left for everybody on this planet. In actuality, population is declining so we need more people on this planet if we are to survive as a species.”

Now we’re getting somewhere: A reference to some facts. There are some interesting points about the population from that posted reference:
– the population is not declining, but the rate of growth is decreasing.
– The annual growth rate peaked at 2.19% in 1963; now, it’s down to 1.15%.
– The UN predicts that growth rate will continue to decline, down to an effective zero growth rate, and that the world population will stabilize at or near 10 Billion by the year 2200.
– Today (2009) the world population is in the neighborhood of 6.8 billion.

So, let’s evaluate the assertion from the first post: “The reason we are so messed up is overpopulation.” Specifically, I wondered if we’d even have enough land to feed all those people. The details I found are below — but the bottom line answer is:
– We’ve got plenty of land now to feed 6.8 Billion people.
– We’ll have to economize a bit to feed 10 Billion.
So it’s not lack of food for the population that is making us “messed up”. Distributing the food is, I think, a problem. Some people have plenty, some have way not enough. Also, the amount of heat and pollution that we currently create, per person – I think that’s a problem, too.

Seems to me that the results of Earth Hour, when we all turn off the lights, is actually a good start. A good next step would be for everyone to turn off their lights more often.

1. 10 Billion people is an interesting figure, considering this statement: “The earth currently produces 2,264 million metric tons of cereals, which is the staple food of the world. If each person consumes 2,000 calories per day, 2,264 million metric tons of cereal will support a little bit over 10 billion people. Currently, around half of all arable land in the world is producing crops.” http://www.aboutmyplanet.com/environment/how-much-human-life-can-planet-earth-sustain/

3. How much land can we grow food on? “About three-quarters of the earth’s surface is covered by water. Of the 57 million or so square miles of land, half are covered by mountains, deserts, or polar ice caps.

That leaves only one-eighth of the earth’s surface for man’s use in farming. And up to the 20th century, about half of that available land has been lost to erosion.

4. By my math, if the needs of 10 Billion people must be supplied by 15 Million square miles, then 667 people must be supported by each square mile. And a square mile is 640 Acres (http://www.google.com/search?hl=en&q=how+many+acres+in+a+square+mile&aq=0&oq=How+many+acres). So each person would be supplied by .96 acres. (Today, with only 6.8 Billion people, each person has 1.4 acres available for food supply.) This is my math, and these are only simple averages. For a reality check, I would suggest that folks in Kansas, say, actually have more useful acres available for food production than folks in, say, the Sahara desert.

5. How much land does it take to feed one person? “The minimum amount of agricultural land necessary for sustainable food security, with a diversified diet similar to those of North America and Western Europe (hence including meat), is 0.5 of a hectare per person … It is realistic to suppose that the absolute minimum of arable land to support one person is a mere 0.07 of a hectare–and this assumes a largely vegetarian diet, no land degradation or water shortages, virtually no post-harvest waste, and farmers who know precisely when and how to plant, fertilize, irrigate, etc. [FAO, 1993]” http://ask.metafilter.com/77287/How-much-land-does-a-person-need

6. My calculations and conclusions: First, that’s an interesting range of estimates — from 0.07 to 0.5 hectares – they differ by a factor of 7. The lower estimate must be really subsistence level (with the upper estimate being fairly upscale!) Back to Google: 1 Acre = .405 Hectares. So that translates to: we need between 0.17 and 1.2 acres per person. For the current population of 6.8 billion, we have more than enough for everyone to eat at the upscale level: we’ve currently got 1.4 acres per person. For the projected stabilized population of 10 Billion, when each person would have (on average) .96 acres available for food, we’ll have to cut back on our lifestyle, but we won’t be near the really-subsistence level of .17 acres per person.

These are publicly available addresses that can be used for communicating between two devices over the network.

What is an IP address?

An IP address is a 48-bit number that identifies one participant in a two-way conversation over the Internet.

How is an IP Address written?

An IP address is written like this: 192.168.1.8 port 80. The valid range of the first four numbers is 0 to 255; the valid range of the last number is 0 to 65,535. This numbering scheme allows for 281,474,976,710,656 unique addresses — however, some are reserved for use by private and multi-cast networks, so only 262,600,608,710,656 are available for public use.

I should note here that most other references that you’ll see to the term “IP Address” do not include the port as part of the address. With that more conventional definition, the number of IP addresses is 232, giving an address count of 4,294,967,296 — that is, a little over 4 billion, rather than the 262 trillion number that I refer to. Why do I use the larger number? In fact, some fairly large part of the internet consists of corporate and institutional networks that use port-numbers to create distinct IP addresses using a technique called Port Address Translation. So that’s why I include the ports in the overall IP Address count.

How is an IP Address used?

When two participants establish a conversation on the Internet, they exchange data in so-called Internet Protocol packets that consist of a prefix (called the “header”) and the data (called the “Payload”). The IP header contains the IP Address of the sender and the IP Address of the recipient for that data packet.

The Internet Protocol is a “hand-shake” agreement between the computers on the Internet that defines how data should be formatted and routed between computers.

I should clarify the term “participant”. I’ll use an example. Right now, my home computer is attached to the Internet and I have 8 windows open to different web-sites:
– A page from Google (The search web-site)
– A page from WikiPedia (The online encyclopedia web-site)
– A page from my web email
– A page from the Washington Conference center
– .. and more.
Each of these pages represents an Internet conversation between my computer and a different computer on the Internet. I’ve got 8 separate conversations, and thus my machine is a participant 8 times — meaning that I have 8 Internet addresses in use right now (for my machine) and 8 external internet addresses reserved for my use, one at each of the external machines that I’m conversing with. Generally, the first 4 numbers of the IP address (192.168.1.8 in my case) identify the machine, and the last number — the port number — identifies the conversation.

Are there enough IP addresses to go around?

The US Population is 303,824,640 (July 2008 est. – from the World Fact Book, published on the web by the CIA). From the US Census bureau, we get these figures, estimated as of today (prepared on February 15, 2009):
U.S. 305,823,230 (305 Million plus) World 6,760,726,213 (6 Billion, 760 Million, plus)

People are concerned that we will eventually run out of IP addresses. So they’ve come up with an extended addressing scheme that will support 2128 (about 3.4×1038) addresses. That is 34 followed 37 additional zeros:
340, 000,000,000, 000,000,000, 000,000,000, 000,000,000

The number name for each triad of 000’s, starting at the left: undecillion, decillion, nonillion, octillion, septillion, sextillion, quintillion, quadrillion, trillion, billion, million, thousands, and hundreds.

So read this as 340 undecillion. Should be enough, eh? (I didn’t include the ports in this calculation! To include the effect of the ports, mutiply 340 undecilion by 65,536. I’ll leave that calcuation as a homework assignment for you.)

Prepared by Bob Monahon, 2/15/2009

A note about the image: This Wikipedia and Wikimedia Commons image is from the user Chris 73 and is freely available at http://commons.wikimedia.org/wiki/File:WorldWideWebAroundGoogle.png under the creative commons cc-by-sa 2.5 license.
]]>https://bobmonahon.wordpress.com/2009/03/07/how-many-ip-addresses-are-there/feed/0Bob Monahon120px-worldwidewebaroundgoogleHello world!https://bobmonahon.wordpress.com/2009/03/01/hello-world/
https://bobmonahon.wordpress.com/2009/03/01/hello-world/#respondSun, 01 Mar 2009 19:26:08 +0000Welcome to BobMonahon.Wordpress.com. So, it’s Sunday and I’m doing my taxes. My daughter Catherine’s in Washington, DC for the Power Shift student demonstration. She’s our young activist! With the big winter storm descending on the East coast tonight and tomorrow, she’ll stay an extra day and head back to school on Tuesday.

I just joined Facebook. The New York Times recently printed that Facebook is becoming the home for baby-boomers to meet their old high-school friends that they’ve completely forgotten. That’s true for me.

I couldn’t get my picture into the blog heading, so it’s here for now. (I also put it on the About page – that’s probably enough.)

I tried the image upload for the customized heading, but the upload/automatic cropping sequence did not work as advertised. After 4 tries with different options, I found I could get a very blurry full face, or a properly focused view of my nose. Since the visual package works best with all the parts showing, I scrapped that plan.

Also, I’d like to make that blog header smaller – no need to take up all that space on the browser page. Couldn’t find an option for that, either. Maybe when I get a chance I’ll try a different theme.